Characterization of Modal Coupling in Nonclassically Damped Systems

1992 ◽  
Author(s):  
F. Ma ◽  
I. W. Park ◽  
J. S. Kim
Keyword(s):  
Large Scale ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Frequency Separation ◽  
Common Procedure ◽  
Damping Matrix ◽  
Natural Modes ◽  
Modal Damping ◽  
Modal Coupling ◽  
Damped Systems

Abstract A common procedure in the solution of a nonclassically damped linear system is to neglect the off-diagonal elements of the associated damping matrix. For a large-scale system, substantial reduction in computational effort is achieved by this method of decoupling the system. Clearly, the decoupling approximation is valid only if modal coupling can somehow be neglected. The purpose of this paper is to study the characteristics of modal coupling, which is amenable to a complex representation. An analytical formulation that facilitates the evaluation of modal coupling is developed. Contrary to widely accepted beliefs, it is shown that neither frequency separation of the natural modes nor strong diagonal dominance of the modal damping matrix would be sufficient to suppress the sometimes significant effect of modal coupling.

Approximate Solution of Nonclassically Damped Systems

1991 ◽  
Author(s):  
F. Ma ◽  
J. H. Hwang
Keyword(s):  
Approximate Solution ◽  
Large Scale ◽  
Computational Effort ◽  
Frequency Separation ◽  
Common Procedure ◽  
Damping Matrix ◽  
Complex Modes ◽  
Modal Coupling ◽  
Order Of Magnitude ◽  
Damped Systems

Abstract In analyzing a nonclassically damped linear system, one common procedure is to neglect those damping terms which are nonclassical, and retain the classical ones. This approach is termed the method of approximate decoupling. For large-scale systems, the computational effort at adopting approximate decoupling is at least an order of magnitude smaller than the method of complex modes. In this paper, the error introduced by approximate decoupling is evaluated. A tight error bound, which can be computed with relative ease, is given for this method of approximate solution. The role that modal coupling plays in the control of error is clarified. If the normalized damping matrix is strongly diagonally dominant, it is shown that adequate frequency separation is not necessary to ensure small errors.

On the Approximate Solution of Nonclassically Damped Linear Systems

1993 ◽  
Vol 60 (3) ◽  
pp. 695-701 ◽  
Author(s):  
J. H. Hwang ◽  
F. Ma
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Error Bound ◽  
Large Scale ◽  
Substantial Reduction ◽  
Approximation Error ◽  
Computational Effort ◽  
Common Procedure ◽  
Damping Matrix ◽  
Modal Damping

A common procedure in the solution of a nonclassically damped linear system is to neglect the off-diagonal elements of the associated modal damping matrix. For a large-scale system, substantial reduction in computational effort is achieved by this method of decoupling the system. In the present paper, the error introduced by disregarding the off-diagonal elements is evaluated, and a quadrature formula for the approximation error is derived. A tight error bound is then obtained. In addition, an effective scheme to improve the accuracy of the approximate solution is outlined.

Characteristics of Modal Coupling in Nonclassically Damped Systems Under Harmonic Excitation

1994 ◽  
Vol 61 (1) ◽  
pp. 77-83 ◽  
Author(s):  
I. W. Park ◽  
J. S. Kim ◽  
F. Ma
Keyword(s):  
Harmonic Excitation ◽  
Frequency Separation ◽  
Diagonal Dominance ◽  
Normal Coordinates ◽  
Damping Matrix ◽  
Damped System ◽  
Natural Modes ◽  
Modal Damping ◽  
Modal Coupling ◽  
Damped Systems

The normal coordinates of a nonclassically damped system are coupled by nonzero off-diagonal elements of the modal damping matrix. The purpose of this paper is to study the characteristics of modal coupling, which is amenable to a complex representation. An analytical formulation is developed to facilitate the evaluation of modal coupling. Contrary to widely accepted beliefs, it is shown that enhancing the diagonal dominance of the modal damping matrix or increasing the frequency separation of the natural modes need not diminish the effect of modal coupling. The effect of modal coupling may even increase. It is demonstrated that, within the practical range of engineering applications, neither diagonal dominance of the modal damping matrix nor frequency separation of the natural modes would be sufficient for neglecting modal coupling.

Diagonal Dominance and the Decoupling Approximation in Damped Discrete Linear Systems

2007 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Diagonal Element ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Damped System ◽  
Modal Damping ◽  
Principal Coordinates ◽  
Discrete Linear Systems ◽  
Modal Coupling ◽  
Decoupling Approximation ◽  
Damped Systems

The principal coordinates of a non-classically damped linear system are coupled by nonzero off-diagonal element of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely accepted that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively believed that the more diagonal the modal damping matrix, the less will be the errors in the decoupling approximation. Two quantitative measures are proposed in this paper to measure the degree of being diagonal dominant in modal damping matrices. It is demonstrated that, over a finite range, errors in the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is presented. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.

Some Remarks About the Decoupling Approximation of Damped Linear Systems

2009 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Linear Systems ◽  
Approximation Error ◽  
Driving Frequency ◽  
Finite Range ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Modal Damping ◽  
Diagonally Dominant ◽  
Decoupling Approximation ◽  
Damped Systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. These unexpected drifts in errors due to the decoupling approximation can be observed at any driving frequency. Small off-diagonal elements in the modal damping matrix may not be sufficient to ensure small errors due to the decoupling approximation. Error-criteria based solely upon diagonal dominance of the modal damping matrix cannot be accurate.

A Study of Errors Induced by Decoupling Approximation in Damped Linear Vibratory Systems

2005 ◽  
Author(s):  
N. Ajavakom ◽  
F. Ma
Keyword(s):  
Coupling Effect ◽  
Vibratory System ◽  
Damping Matrix ◽  
Modal Damping ◽  
Principal Coordinates ◽  
Decoupling Approximation ◽  
Damped Systems

It is well known that an undamped linear vibratory system can be decoupled through transformation to principal coordinates. In the presence of damping, coordinate decoupling occurs only if the system is classically damped. Upon modal transformation, the system generally remains coupled by the off-diagonal elements of its modal damping matrix. A common approximation in the analysis of nonclassically damped systems is to ignore the off-diagonal elements of the modal damping matrix, which is equivalent to neglecting coupling of the principal coordinates. This procedure is termed the decoupling approximation. Intuitively, the errors of decoupling approximation should be small if the off-diagonal elements of the modal damping matrix are small. Contrary to this widely accepted belief, an example is provided to demonstrate that this criterion is not sufficient for decoupling approximation. In fact, coupling effect can even increase as the off-diagonal elements of the modal damping matrix decrease in magnitude. Discussion and explanation are provided as to why the errors increase when the modal damping matrix becomes increasingly diagonal.

scholarly journals Computational Issues in Damping Identification for Large Scale Problems

1997 ◽  
Author(s):  
Deborah F. Pilkey ◽  
Kevin P. Roe ◽  
Daniel J. Inman
Keyword(s):  
High Performance ◽  
Large Scale ◽  
Least Squares Method ◽  
Diagnostic Techniques ◽  
Computational Effort ◽  
Analytical Models ◽  
High Performance Fortran ◽  
Damping Matrix ◽  
Large Scale Problems ◽  
Vibration Responses

Abstract Damage detection and diagnostic techniques using vibration responses that depend on analytical models provide more information about a structure’s integrity than those that are not model based. The drawback of these approaches is that some form of a workable model is required. Typically, models of practical structures and their corresponding computational effort are very large. One method of detecting damage in a structure is to measure excess energy dissipation, which can be seen in damping matrices. Calculating damping matrices is important because there is a correspondence between a change in the damping matrix and the change in the health of a structure. The objective of this research is to investigate the numerical problems associated with computing damping matrices using inverse methods. Two damping identification methods are tested for efficiency in large-scale applications. One is an iterative routine, and the other a least squares method. Numerical simulations have been performed on multiple degree-of-freedom models to test the effectiveness of the algorithm and the usefulness of parallel computation for the problems. High Performance Fortran is used to parallelize the algorithm.

Identification of Multi-Degree of Freedom Systems With Nonproportional Damping Using the Resonant Decay Method

2004 ◽  
Vol 126 (2) ◽  
pp. 298-306 ◽  
Author(s):  
Steven Naylor ◽  
Michael F. Platten ◽  
Jan R. Wright ◽  
Jonathan E. Cooper
Keyword(s):  
Measurement Noise ◽  
Mode Shapes ◽  
Damping Matrix ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Rigid Body Modes ◽  
Nonproportional Damping ◽  
Modal Mass ◽  
Damped Systems ◽  
Modal Space

This paper describes an extension of the force appropriation approach which permits the identification of the modal mass, damping and stiffness matrices of nonproportionally damped systems using multiple exciters. Appropriated excitation bursts are applied to the system at each natural frequency, followed by a regression analysis in modal space. The approach is illustrated on a simulated model of a plate with discrete dampers positioned to introduce significant damping nonproportionality. The influence of out-of-band flexible and rigid body modes, imperfect appropriation, measurement noise and impure mode shapes is considered. The method is shown to provide adequate estimates of the modal damping matrix.

Study on Equivalent Damping Ratio of Shenzhen Baoan International Airport T3 Terminal

2012 ◽  
Vol 446-449 ◽  
pp. 871-877
Author(s):  
Yu Chen Yang ◽  
Lei Gu ◽  
Zhong Yi Zhu ◽  
Kai Qin ◽  
Lin Zhang
Keyword(s):  
Modal Analysis ◽  
Damping Ratio ◽  
Earthquake Response ◽  
Structural System ◽  
Equivalent Damping ◽  
Damping Matrix ◽  
Modal Damping ◽  
Modal Coupling ◽  
Response Equation ◽  
Energy Theory

Today, the structures constituted by different materials, increase more and more, especially the lower part is concrete, the upper is steel. For this type of structural system, modal damping matrix is non-diagonal matrix, the earthquake response equation is coupled on the modal damping matrix, the modal coupling of the non-proportional damping system leads to the traditional real modal analysis methods not be directly applied. For such structure, the changes of the damping ratio are analyzed in this article. Finally, the equivalent damping ratio of Shenzhen Airport is obtained using the energy theory.

scholarly journals On Decoupling the Equations of Motion of Nonclassically Damped Systems

1989 ◽  
Author(s):  
F. Ma ◽  
J. H. Hwang
Keyword(s):  
Approximate Solution ◽  
Linear Systems ◽  
Error Bounds ◽  
Equations Of Motion ◽  
Effective Procedure ◽  
Common Procedure ◽  
Damping Matrix ◽  
Alternative Techniques ◽  
Damped Systems

One common procedure in the solution of a damped linear systems with small off-diagonal damping elements is to neglect the off-diagonal elements of the normalized damping matrix. The extent of approximation introduced by this method of decoupling the system is evaluated, and tight error bounds are derived by alternative techniques. An effective procedure to improve the accuracy of the approximate solution is outlined.

Sign in / Sign up

Export Citation Format

Share Document